\subsubsection{Wrench Space}
Where the CMCCP quality measure relies on geometric properties only, the Wrench Space quality measure takes the amount of force and torque applied by the gripper into account.\\

\noindent For each contact point, the normal to the contact point is given along with the force applied by the gripper. In order to take friction into account, a friction cone approximation is used. The basic idea is that the contact force is known to lie within a cone. The width of this cone is limited by the force that the friction between the gripper and the object can resist. Figure \ref{fig:cone:1} shows this.

\begin{figure}[H]
\centering
\includegraphics[width = 0.25\textwidth]{figures/cone.png}
\caption{Friction Cone.}
\label{fig:cone:1}
\end{figure}

\noindent The friction cone can be described as in the following equation for each contact point.
\begin{equation}
||f_i-(f_i \cdot n_i)n_i|| \le -\mu (f_i \cdot n_i)
\end{equation}
where $f_i$ depicts the force that can be applied in a contact point, $\mu$ the friction coefficient and $n_i$ the contact normal in contact point $i$, as described in \cite{GraspPlanning}.\\
In reality the friction cone must be approximated. This is done by using the normal vector of the contact. The maximum frictional component is then calculated, and this vector is then rotated around the contact normal force, in steps according to the resolution that is wanted. Obviously the resolution should be chosen as high as possible to give the best approximation. A high resolution does unfortunately affect the complexity of the convex hull calculations, and therefore the resolution should not be chosen too high.\\

\noindent Rather than using only the forces to build a 3D wrench space, the torques should be included as well, resulting in a 6D wrench space. All forces acting on the object (including those from the friction cone) will result in a torque on the object. The torques must be in equilibrium as well as the forces, in order for the object to be in rest.

\noindent As a result of the forces, the following torques are created.
\begin{equation}
\tau_i = (c_i-r)\times f_i
\end{equation}
where $c_i$ denotes the center of mass, and r is the position of the contact point.\\

\noindent Given the force and the torque, a wrench can be written as the combination of both.
\begin{equation}
w_i=\begin{pmatrix}
f_i\\
(c_i-r)\times f_i
\end{pmatrix}
\end{equation}

\noindent Given the wrenches from all contact points, a convex hull can be created. Figure \ref{fig:cone} illustrates how this works in practice. In figure \ref{fig:cone:2} a 2D case is illustrated to show the basic principle of the force part of the wrench space. For each of the contact points a friction cone is calculated. Here the friction cones indicate the forces the gripper is able to resist in the given contact points. The convex hull created from this is shown in figure \ref{fig:cone:3}. The grasp will be able to resist any wrench that can fit inside the convex hull. For simplicity this convex hull is only shown in 2 dimensions (for the forces). In reality it will be a 6-dimensional space.
\begin{figure}[H]
\centering
\subfigure[Object and Contacts.]{\includegraphics[width = 0.40\textwidth]{figures/objectWCones.png}\label{fig:cone:2}}
\hspace{1cm}
\subfigure[Convex Hull.]{\includegraphics[width = 0.15\textwidth]{figures/convex.png}\label{fig:cone:3}}
\caption{Object with 3 contacts, and the convex hull.}
\label{fig:cone}
\end{figure}

\noindent Based on the constructed convex hull a quality measure can be based on the minimum distance to the hull from the reference point, depicting the minimum amount of force and torque that the grasp can withstand in any direction. A 2-norm is used to find the size of this minimum distance. As force and torque use two different units that does not necessarily have comparable sizes, a scaling factor is used. The default scaling for the torque in the RobWork Simulator is
\begin{equation}
\lambda _\tau = \frac{1}{\mbox{MaxDistance}}
\label{eq:lambda:torque}
\end{equation}
where MaxDistance is the maximum distance from any point on the object to the center of mass. This factor makes sure that the quality measure is independent of object scale.
\clearpage
\noindent An alternative measure could be based on a normalized friction cone, where the forces magnitude are normalized. If then the centre point is inside the convex hull the grasp is supposedly good. As the interest lies on being able to compare the quality measure with actual forces, this is though not investigated further.\\